Estimating the True Distribution of Data Collected with Randomized Response

2026-01-13

AI Summary

This work addresses a key limitation of the standard debiasing approach for the Randomized Response (RR) mechanism under local differential privacy, which often yields negative estimates that lack interpretability as histogram probabilities. The paper presents the first closed-form maximum likelihood estimator (MLE) for the true underlying distribution under RR, circumventing the high computational cost of iterative Bayesian updating (IBU) algorithms. The proposed estimator is theoretically elegant, computationally efficient, and guarantees non-negative, accurate reconstruction of the original histogram. Empirical evaluations demonstrate that this analytical solution significantly outperforms existing correction methods in both estimation accuracy and computational efficiency, offering practitioners a reliable and scalable tool for privacy-preserving frequency estimation.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Randomized Response (RR) is a protocol designed to collect and analyze categorical data with local differential privacy guarantees. It has been used as a building block of mechanisms deployed by Big tech companies to collect app or web users'data. Each user reports an automatic random alteration of their true value to the analytics server, which then estimates the histogram of the true unseen values of all users using a debiasing rule to compensate for the added randomness. A known issue is that the standard debiasing rule can yield a vector with negative values (which can not be interpreted as a histogram), and there is no consensus on the best fix. An elegant but slow solution is the Iterative Bayesian Update algorithm (IBU), which converges to the Maximum Likelihood Estimate (MLE) as the number of iterations goes to infinity. This paper bypasses IBU by providing a simple formula for the exact MLE of RR and compares it with other estimation methods experimentally to help practitioners decide which one to use.

Problem

• Randomized Response
• Local Differential Privacy
• Histogram Estimation
• Negative Estimates
• Maximum Likelihood Estimate

Innovation

• Randomized Response
• Maximum Likelihood Estimation
• Local Differential Privacy
• Histogram Estimation
• Closed-form Solution